   # Chebyshev’sEquation ConsiderChebyshev’s equation ( 1 − x 2 ) y ″ − x y ′ + k 2 y = 0 Polynomial solutions of this differential equation are called Chebyshev polynomials and are denoted by T k (x). They satisfy the recursion equation T n + 1 ( x ) = 2 x T n ( x ) − T n − 1 ( x ) . Given that T 0 ( x ) = 1 and T 1 ( x ) = x , determine the Chebyshevpolynomials T 2 ( x ) , T 3 ( x ) and T 4 ( x ) . Verify that T 0 ( x ) , T 1 ( x ) , T 2 ( x ) , T 3 ( x ) , and T 4 ( x ) are solutions of the given differential equation. Verify the following Chebyshev polynomials. T 5 ( x ) = 16 x 5 − 20 x 3 + 5 x T 6 ( x ) = 32 x 6 − 48 x 4 + 18 x 2 − 1 T 7 ( x ) = 64 x 7 − 112 x 5 + 56 x 3 − 7 x ### Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378

#### Solutions

Chapter
Section ### Multivariable Calculus

11th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337275378
Chapter 16, Problem 16PS
Textbook Problem
1 views

## Chebyshev’sEquation ConsiderChebyshev’s equation ( 1 − x 2 ) y ″ − x y ′ + k 2 y = 0 Polynomial solutions of this differential equation are called Chebyshev polynomials and are denoted by Tk(x). They satisfy the recursion equation T n + 1 ( x ) = 2 x T n ( x ) − T n − 1 ( x ) .Given that T 0 ( x ) = 1   and   T 1 ( x ) = x , determine the Chebyshevpolynomials T 2 ( x ) , T 3 ( x ) and T 4 ( x ) .Verify that T 0 ( x ) , T 1 ( x ) , T 2 ( x ) , T 3 ( x ) , and   T 4 ( x ) are solutions of the given differential equation.Verify the following Chebyshev polynomials. T 5 ( x ) = 16 x 5 − 20 x 3 + 5 x T 6 ( x ) = 32 x 6 − 48 x 4 + 18 x 2 − 1 T 7 ( x ) = 64 x 7 − 112 x 5 + 56 x 3 − 7 x

(a)

To determine

To calculate: The value of Chebyshev polynomials T2(x),T3(x),andT4(x) where T0(x)=1andT1(x)=x.

### Explanation of Solution

Given information: TheChebyshev’sequation (1x2)yxy+k2y=0.

The recursion equation Tn+1(x)=2xTn(x)Tn1(x).

T0(x)=1andT1(x)=x.

Calculation:

Substitute n=1 in the recursion equation Tn+1(x)=2xTn(x)Tn1(x) to get T2(x)=2xT1(x)T0(x).

Now, T0(x)=1andT1(x)=x in the above formula to get the value of T2(x).

T2(x)=2xT1(x)T0(x)T2(x)=2x.x1T2(x)=2x21

Substitute n=2 in the recursion equation Tn+1(x)=2xTn(x)Tn1(x) to get T3(x)=2xT2(x)T1(x).

Now, T1(x)=xandT2(x)=2x21 in the above formula to get the value of T3(x)

(b)

To determine

To prove: That T0(x),T1(x),T2(x),T3(x),andT4(x) are the solutions of the differential equation (1x2)yxy+k2y=0.

(c)

To determine

To prove: T5(x)=16x520x3+5x,T6(x)=32x648x4+18x21,and T7(x)=64x7112x5+56x37x.

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find more solutions based on key concepts
. ∞ −∞ 1 −1

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Find all possible real solutions of each equation in Exercises 3144. y3+3y2+3y+2=0

Finite Mathematics and Applied Calculus (MindTap Course List)

Determine whether the series is convergent or divergent. 23. k=1kek

Single Variable Calculus: Early Transcendentals

In Exercises 1722, sketch the graph of the function f and evaluate limxaf(x), if it exists, for the given value...

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Find each product. 18rs(3s16t)

Elementary Technical Mathematics

ReminderRound all answers to two decimal places unless otherwise indicated. World Population from Two Points On...

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)

The Goodman Tire and Rubber Company periodically tests its tires for tread wear under simulated road conditions...

Modern Business Statistics with Microsoft Office Excel (with XLSTAT Education Edition Printed Access Card) (MindTap Course List)

The graph of a solution of a second-order initial-value problem d2y/dx2 = f(x, y, y), y(2) = y0, y(2) = y1, is ...

A First Course in Differential Equations with Modeling Applications (MindTap Course List) 